MATH 392 - Topics in Abstract Algebra

Credits: 3 Prerequisite: MATH 321. Topics vary but can include field and Galois theory, geometric and combinatorial group theory, representation theory, number theory, algebraic number theory, commutative algebra, algebraic geometry, arithmetic geometry, advanced linear algebra, algebraic coding theory and cryptography, algebraic topology, homological algebra, and graph theory, May be repeated for degree credit if the topic is different.

Winter 2021, MATH 392A-01: Topics in Abstract Algebra: Algebraic Number Theory (3). Prerequisite: MATH 321. Number theory studies questions about the integers (among other things). As an example, the equation x^2 + y^2 = z^2 defines a cone in three-dimensional space. A number theorist might ask for the integer triples (x,y,z) that satisfy this equation. These are also called Pythagorean triples since (ignoring signs) they arise as the side lengths of right-angled triangles. You may be aware that (3,4,5) and (5,12,13) are such triples. Are there others? Can they be described in a systematic fashion? What happens if we change the equation? In this course, we’ll see how these sorts of questions can be addressed. Ideas and tools from algebra such as modular arithmetic and the notion of unique factorization in various systems will play central roles. Bush.